Men of Mathematics (87 page)

“But, having once arrived at this point of view, the problems—vast enough—which I had thought to propose to myself, seemed inconsiderable beside the great questions of the general theory of forms. In this boundless expanse of researches which Monsieur Gauss [Gauss was still living when Hermite wrote this, hence the polite “Monsieur”]
has opened up to us, Algebra and the Theory of Numbers seem necessarily to be merged in the same order of analytical concepts, of which our present knowledge does not yet permit us to form an accurate idea.”

He then makes a remark which, although not very clear, can be interpreted as meaning that the key to the subtle connections between algebra, the higher arithmetic, and certain parts of the theory of functions will be found in a thorough understanding of
what sort
of “numbers” are both necessary and sufficient for the explicit solution of all types of algebraic equations. Thus, for
x
³
−1 = 0, it is necessary and sufficient to understand
for
x
⁵
+ ax
+
b =
0, where
a, b
are any given numbers, what sort of a “number”
x
must be invented in order that
x
may be expressed
explicitly
in terms of
a, b?
Gauss of course gave one kind of answer: any root
x
is a complex number. But this is only a beginning. Abel proved that if only a
finite
number of rational operations and extractions of roots are permitted, then there is
no
explicit formula giving
x
in terms of
a, b.
We shall return to this question later; Hermite even at this early date
(1848;
he was then twenty six) seems to have had one of his greatest discoveries somewhere at the back of his head.

In his attitude toward numbers Hermite was somewhat of a mystic in the tradition of Pythagoras and Descartes—the latter's mathematical creed, as will appear in a moment, was essentially Pythagorean. In other matters, too, the gentle Hermite exhibited a marked leaning toward mysticism. Up to the age of forty three he was a tolerant agnostic, like so many French men of science of his time. Then, in
1856,
he fell suddenly and dangerously ill. In this debilitated condition he was no match for even the least persistent evangelist, and the ardent Cauchy, who had always deplored his brilliant young friend's open-mindedness on religious matters, pounced on the prostrate Hermite and converted him to Roman Catholicism. Thenceforth Hermite was a devout Catholic, and the practice of his religion gave him much satisfaction.

Hermite's number-mysticism is harmless enough and it is one of those personal things on which argument is futile. Briefly, Hermite believed that numbers have an existence of their own above all control by human beings. Mathematicians, he thought, are permitted now and then to catch glimpses of the superhuman harmonies regulating this ethereal realm of numerical existence, just as the great geniuses
of ethics and morals have sometimes claimed to have visioned the celestial perfections of the Kingdom of Heaven.

It is probably right to say that no reputable mathematician today who has paid any attention to what has been done in the past fifty years (especially the last twenty five) in attempting to understand the nature of mathematics and the processes of mathematical reasoning would agree with the mystical Hermite. Whether this modern skepticism regarding the other-worldliness of mathematics is a gain or a loss over Hermite's creed must be left to the taste of the reader. What is now almost universally held by competent judges to be the wrong view of “mathematical existence” was so admirably expressed by Descartes in his theory of the eternal triangle that it may be quoted here as an epitome of Hermite's mystical beliefs.

“I imagine a triangle, although perhaps such a figure does not exist and never has existed anywhere in the world outside my thought. Nevertheless this figure has a certain nature, or form, or determinate essence which is immutable or eternal, which I have not invented and which in no way depends on my mind. This is evident from the fact that I can demonstrate various properties of this triangle, for example that the sum of its three interior angles is equal to two right angles, that the greatest angle is opposite the greatest side, and so forth. Whether I desire to or not, I recognize very clearly and convincingly that these properties are in the triangle although I have never thought about them before, and even if this is the first time I have imagined a triangle. Nevertheless no one can say that I have invented or imagined them.” Transposed to such simple “eternal verities” as 1 + 2 = 3, 2 + 2 = 4, Descartes' everlasting geometry becomes Hermite's superhuman arithmetic.

One arithmetical investigation of Hermite's, although rather technical, may be mentioned here as an example of the prophetic aspect of pure mathematics. Gauss, we recall, introduced
complex integers
(numbers of the form
a
+
bi,
where
a, b
are rational integers and
i
denotes
into the higher arithmetic in order to give the law of biquadratic reciprocity its simplest expression. Dirichlet and other followers of Gauss then discussed quadratic forms in which the rational integers appearing as variables and coefficients are replaced by Gaussian complex integers. Hermite passed to the general case of this situation and investigated the representation of integers in what are today called
Hermitian forms.
An example of such a form (for the
special case of two complex variables
x
₁
,
x
₂
and their “conjugates”
instead of
n
variables) is

in which the bar over a letter denoting a complex number indicates the
conjugate
of that number; namely, if
x
+
iy
is the complex number, its “conjugate” is
x—iy;
and the coefficients
a
₁₁
, a
₁₂
, a
₂₁
a
₂₂
are such that
a
_ij
= ā
_ji
, for (
i, j
) = (1, 1), (1, 2), (2, 1), (2, 2), so that
a
₁₂
and
a
₂₁
are conjugates, and each of
a
₁₁
, a
₂₂
is its own conjugate (so that
a
₁₁
, a
₂₂
are real numbers). It is easily seen that the entire form is real (free of
i)
if all products are multiplied out, but it is most “naturally” discussed in the shape given.

When Hermite invented such forms he was interested in finding what numbers are represented by the forms. Over seventy years later it was found that the algebra of Hermitian forms is indispensable in mathematical physics, particularly in the modern quantum theory. Hermite had no idea that his pure mathematics would prove valuable in science long after his death-—indeed, like Archimedes, he never seemed to care much for the scientific applications of mathematics. But the fact that Hermite's work has given physics a useful tool is perhaps another argument favoring the side that believes mathematicians best justify their abstract existence when left to their own inscrutable devices.

Leaving aside Hermite's splendid discoveries in the theory of algebraic invariants as too technical for discussion here, we shall pass on in a moment to two of his most spectacular achievements in other fields. The high esteem in which Hermite's work in invariants was held by his contemporaries may however be indicated by Sylvester's characteristic remark that “Cayley, Hermite, and I constitute an In-variantive Trinity.” Who was who in this astounding trinity Sylvester omitted to state; but perhaps this oversight is immaterial, as each member of such a trefoil would be capable of transforming himself into himself or into either of his coinvariantive beings.

*  *  *

The two fields in which Hermite found what are perhaps the most striking individual results in all his beautiful work are those of the general equation of the fifth degree and transcendental numbers. The nature of what he found in the first is clearly indicated in the introduction to his short note
Sur la rèsolution de lèquation du cinquième degré
(On the Solution of the [general] Equation of the Fifth Degree; published in the
Comptes rendus de l‘Académie des Sciences
for
1858,
when Hermite was thirty six).

“It is known that the general equation of the fifth degree can be reduced, by a substitution [on the unknown x] whose coefficients are determined without using any irrationalities other than square roots or cube roots, to the form

x
⁵
—x—
a
= 0.

[That is,
if
we can solve
this
equation for x,
then
we can solve the general equation of the fifth degree.]

“This remarkable result, due to the English mathematician Jerrard, is the most important step that has been taken in the algebraic theory of equations of the fifth degree since Abel proved that a solution by radicals is impossible. This impossibility shows in fact the necessity for introducing some new analytic element [some new kind of function] in seeking the solution, and, on this account, it seems natural to take as an auxiliary the roots of the very simple equation we have just mentioned. Nevertheless, in order to legitimize its use rigorously as an essential element in the solution of the general equation, it remains to see if this simplicity of form actually permits us to arrive at some idea of the nature of its roots, to grasp what is peculiar and essential in the mode of existence of these quantities, of which nothing is known beyond the fact that they are not expressible by radicals.

“Now it is very remarkable that Jerrard's equation lends itself with the greatest ease to this research, and is, in the sense which we shall explain, susceptible of an actual analytic solution. For we may indeed conceive the question of the algebraic solution of equations from a point of view different from that which for long has been indicated by the solution of equations of the first four degrees, and to which we are especially committed.